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Sealed Kurepa Trees

Itamar Giron, Yair Hayut

Abstract

In this paper we investigate the problem of the distributivity of Kurepa trees. We show that it is consistent that there are Kurepa trees and for every Kurepa tree there is a small forcing notion which adds a branch to it without collapsing cardinals. On the other hand, we derive a proper forcing notion for making an arbitrary Kurepa tree into a non-distributive tree without collapsing $\aleph_1$ and $\aleph_2$.

Sealed Kurepa Trees

Abstract

In this paper we investigate the problem of the distributivity of Kurepa trees. We show that it is consistent that there are Kurepa trees and for every Kurepa tree there is a small forcing notion which adds a branch to it without collapsing cardinals. On the other hand, we derive a proper forcing notion for making an arbitrary Kurepa tree into a non-distributive tree without collapsing and .
Paper Structure (9 sections, 34 theorems, 96 equations)

This paper contains 9 sections, 34 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properness and side conditions
  4. Every Kurepa tree is not-sealed
  5. Distributivity of Solovay's Tree
  6. A model without sealed trees
  7. The Sealing Poset
  8. The Generic Object
  9. Open Questions

Key Result

Theorem 1

Let $T$ be a Kurepa tree then there is a proper forcing notion that makes $T$ non-distributive, and preserves $\aleph_1$ and $\aleph_2$.

Theorems & Definitions (90)

  • Theorem
  • Lemma 2.2
  • proof
  • Theorem 2.4: Baumgartner
  • Theorem 2.6: Solovay JechSolovay
  • Lemma 2.10
  • proof
  • Corollary 2.11
  • Theorem 2.12: Shelah and Poór Poor
  • Corollary 2.13
  • ...and 80 more